『学習院大学 経済論集』第49巻 第2号（2012年7月）
Strategic commitment to pursue a goal other than profit in
a Cournot duopoly
Dimitry Rtischev*
Abstract
Competition among profit-seeking firms in an oligopolistic industry inherently generates incentives
for firms to commit to maximize a performance metric other than profit. We briefly review the
underlying theory, analyze its ramifications in a Cournot duopoly, and consider feasibility constraints
from the perspective of strategic management.
Keywords: oligopolistic competition, strategic commitment, strategic delegation
JEL classification: D43, L13, L21
1. Introduction
If all other players in a strategic interaction are payoff-maximizers, a player can increase its payoff by
committing to maximize something other than its payoff. This paradoxical proposition was proved for a
broad class of games by Heifetz et. al. in their provocatively titled paper “What to maximize if you
must?” (2007) The proposition implies that, if profit-seeking players are allowed to choose what to
maximize, they generally will not choose to maximize profit. This finding brings into question the
validity of many applications of game theory which neglect to consider the possibility that players may
commit to pursue something other than the payoffs. In the context of strategic management, this implies
that we cannot presume on a priori grounds that firms in an oligopolistic industry explicitly and directly
pursue profits. Rather, we should expect firms in an oligopoly to commit to pursue revenue, market
share, or some other performance metric. We should also expect such firms to exert effort to make their
commitments credible and observable.
The argument for why firms in an oligopoly have incentive to commit to maximize something other
than profit has been reviewed by Fershtman and Judd (1987), who also analyzed the ramifications using
a model of Cournot duopoly. Since then, strategic deviations from profit maximization have been studied
in the context of Cournot oligopoly by Blinder (1993), Dufwenberg and Guth (1999), Gehrig et. al.
＊） Faculty of Economics, Gakushuin University, Tokyo, Japan
133
(2004), Miller and Pazgal (2002) and others. The goal of this paper is to simplify the earlier findings to
the bare essentials, clarify the key issues, and qualify the argument by considering feasibility constraints
from a management perspective.
In the next section we will informally review why small-numbers competition among profit-seeking
firms inherently generates incentives for them to commit to pursue an objective other than profit. Section
3 will apply the theory to a Cournot duopoly with constant unit costs, the simplest context that exposes
the key issues. We will then narrow the focus by introducing feasibility constraints in Section 4 and
further discuss feasibility from the management perspective in the concluding Section 5.
2. Why maximize something other than profit
A credible commitment by a player in a strategic interaction can influence the behavior of another
player in a way that benefits the first player. A firm that builds a large and efficient factory makes a
commitment to manufacture more products at a lower cost, and this may induce a competing firm to
cancel plans to build a large factory of its own, resigning to a smaller share of the market, or perhaps
even exit the market. A much less tangible and often overlooked form of commitment is the adoption of
a management philosophy to pursue some performance metric other than profit, for example, revenue or
market share. Just like building a factory, a commitment to a management philosophy can also influence
the behavior of rivals in a way that benefits the firm that makes the commitment. In equilibrium, all firms
may commit to management philosophies and the resulting effect on profits may be positive or negative.
A number of general treatments have examined endogenous strategic choice of objective function. The
conclusions are broad, general, and have support in evolutionary theory. Heifetz, et. al. (2007) formally
demonstrated that players in a generic game have incentive to commit to maximize something other than
the payoffs of the game. Moreover, they showed that such commitments do not disappear under
evolutionary dynamics. Evolution of preferences theory too has shown that agents who maximize a
“subjective utility” different from actual payoffs can evolve and displace agents who maximize actual
payoffs (Guth and Kliemt, 1998). Winter et. al. (2009) corroborated these findings in their analysis of
“mental equilibria.” A critical assumption underlying all these results is that players’ commitments are
credible and can be observed by other players with enough precision. We will return to the issue of
commitment in later sections.
These results imply that there are no a priori grounds for presuming that optimizing entities engaged
in strategic interactions do best by pursuing payoffs as directly and objectively as possible. On the
contrary, the results suggest we should expect optimizing entities in strategic interactions to maximize
something other than what they ultimately seek, and, moreover, strive to credibly communicate their
commitment to maximize that something else.
3. Commitment to a performance metric in Cournot duopoly
Consider a Cournot duopoly facing linear demand p = a – q1– q2, where p is the market-clearing
price, a
i {1, 2} Firms
> 0 is a demand parameter, and qi is the quantity of output produced by firm .
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Strategic commitment to pursue a goal other than profit in a Cournot duopoly (Rtischev)
have constant unit costs ci. We assume 0 ≤ c1 ≤ c2 and a > 2c2 c1 , which ensures that both firms
produce positive quantities in the Cournot-Nash equilibrium. Each firm may adopt a “management
philosophy” that observably and credibly commits it to maximize a specific performance metric. All
quantity produced by the duopoly is sold at the market-clearing price, giving firms revenue
Ri = (a qi q j ) qi and profit i = (a qi q j ci )qi . All of the above is common knowledge.
We will study equilibrium choice of management philosophies and their effects on profitability using
the following two-stage model. In Stage I, firms simultaneously, publicly, and credibly commit to a
management philosophy. Specifically, firm i commits that in Stage II it will maximize a “performance
E
E
function” i (qi , q j , ci , c j , a), where q j is the conjecture by firm i about the quantity to be produced by
firm j. Stage II is standard Cournot competition except that each firm seeks to maximize its chosen
performance function rather than profit. That is, in Stage II firms simultaneously choose quantities qi per
qi* = arg max qi
i
(qi , q Ej , ci , c j , a)
E
*
and in equilibrium conjectures about rivals’ output are fulfilled: qi = qi .
i
The baseline case is when both firms commit to simply maximize profit. Thus, if both firms choose
= i in Stage I, standard Cournot-Nash equilibrium quantities and profits result in Stage II:
Equilibrium A (Standard Cournot-Nash)
qi* = 1 (a 2ci + c j )
3
*
i
1
9
=
(a 2ci + c j )2
Next suppose each firm may commit to estimate demand optimistically or pessimistically by a bias
That is, the performance function firms commit to maximize is:
i
= (a +
i
.
i
qi q Ej ci )qi
Maximization of this performance function in Stage II by both firms gives the equilibrium quantities as a
function of the biases:
qi* ( i ,
j
) = 1 (a 2ci + c j + 2
3
i
j
)
In Stage I each firm looks ahead to Stage II and chooses the bias that will give it the most profit,
assuming the rival does the same:
*
i
= arg max
i
i
(qi*, q *j )
The biases chosen in equilibrium of Stage I are
*
i
= 1 (a 3ci + 2c j )
5
and the resulting quantities and profits are:
135
Equilibrium B
qi* = 2 (a 3ci + 2c j )
5
*
i
=
2
25
(a 3ci + 2c j )2
Equilibrium B also results when firms commit to use optimistic or conservative estimates of their
ci
costs. Specifically, if each firm estimates its unit cost as i , then the performance function is again
E
=
(a
+
q
q
c
)q
and
therefore
the
equilibrium
biases,
quantities, and profits are also the same
i
i
i
j
i
i
as above.
Equilibrium B also results when firms commit to pursue both profit and output. Specifically, if each
firm commits to maximize i = i + i qi, then the performance function again takes the form
qi q Ej ci )qi and therefore the equilibrium biases, quantities, and profits are the same as
i = (a +
i
above.
Equilibrium B also results when firms commit to pursue both profit and revenue. Specifically, if each
firm commits to maximize i = i + i Ri, then the performance function takes the form
i
= ((1
i
)(a qi q Ej ) ci )qi
Maximization of this performance function in Stage II by both firms gives the equilibrium quantities as a
function of the biases:
qi* ( i ,
j
)=
cj
1
2ci
a
+
1+ i 1+
3
2
Optimization in Stage I to maximize profit gives the equilibrium biases:
*
i
=
a 3ci + 2c j
a 8ci + 2c j
It is straightforward to confirm that the resulting quantities and profits are the same as in Equilibrium B.
As summarized in Table 1, several different management philosophies result in Equilibrium B. In
general, a sufficient condition for a management philosophy to result in Equilibrium B is for the
performance function to have the form i = i + i fi (qi ,q Ej ,a,ci ,c j ), where the function fi is such that the
first-order condition
i / qi =
i fi / qi can be expressed as
i / qi = g( i ,a,ci ,c j ) for some wellbehaved function g. An example of a management philosophy which does not meet this criterion is a
performance function that incorporates concern for market share: θi = π i + ω i qi /(qi + q j ). This leads to an
essentially different maximization problem whose equilibrium is much less tractable.
Comparing firms’ performance in Equilibrium A and Equilibrium B reveals that the adoption of
management philosophies results in:
▪ more output by the more efficient firm 1
▪ more or less output by the less efficient firm 2
▪ possibility of firm 2 shutting down
▪ more industry output and lower market price
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Strategic commitment to pursue a goal other than profit in a Cournot duopoly (Rtischev)
Table 1. Some management philosophies that result in Equilibrium B. It is assumed that
parameters are restricted to ranges that yield interior equilibria.
Management philosophy
Demand optimism or
pessimism
Cost optimism or
pessimism
Concern for profit and
quantity
Concern for profit and
revenue
Performance function
i
= i with a
a+
i
= i with ci
ci
i
=
i
+
i i
i
=
i
+
i
Equilibrium bias
*
i
i
= 1 (a 3ci + 2c j )
5
’’
i
q
’’
*
i
Ri
=
a 3ci + 2c j
a 8ci + 2c j
▪ higher or lower profit for firm 1
▪ lower or zero profit for firm 2
Overall, management philosophies intensify competition and may even hurt the profit of the more
efficient firm. These conclusions do not critically depend on the assumption that the firms’ products are
prefect substitutes. An n-firm oligopoly model that allows for various degrees of strategic substitutability
or complementarity reached qualitatively similar conclusions (Gehrig et. al., 2004) In that model, each
firm could commit to use a biased estimate of the degree of substitutability or complementarity of its
product vis-à-vis products sold in the same market by other firms. In equilibrium, firms committed to
over-estimate substitutability or under-estimate complementarity, leading them to compete more
aggressively in the output market and earn lower profits.
Our assumption of constant unit costs is also not critical to the overall conclusions. In a Cournot
duopoly where costs increase quadratically with output, Dufwenberg and Guth (1999) allowed firms to
commit to maximize a linear combination of profit and output and found that in equilibrium firms do
make such commitments and end up competing more aggressively, resulting in lower price, higher
output, and lower profits.
Lastly, our conclusions do not critically depend on the assumption of certainty about costs and
demand. Fershtman and Judd (1987) allowed firms to commit to maximize a linear combination of profit
and revenue in a Cournot duopoly under conditions of cost and demand uncertainty. In the equilibria of
their model, firms commit to place some weight on revenue rather than just pursue profit and end up
competing more aggressively, driving down both price and profits.
4. Feasible commitments in Cournot duopoly: profit or revenue
We have been tacitly assuming that firms can fine-tune their management philosophies by choosing
from a large space of performance functions parameterized by
. Considering that firms must not only
i
choose a performance function but also explain and commit to the corresponding management
137
philosophy both internally and externally, and then actually implement it within the organization, the set
of feasible performance functions is probably very limited. In the context Cournot oligopoly, the two
simplest and therefore most feasible performance functions are profit and revenue. Therefore, we will
next restrict choice of management philosophies to either profit or revenue. Specifically, we will study
the following simplified version of two-stage game that we introduced in the previous section:
Stage I: each firm simultaneously, publicly, and credibly chooses to commit to maximize either Profit
{ i ,Ri }
Stage II: each firm simultaneously chooses a quantity that maximizes the performance metric it
selected in Stage I, taking into account that its rival is also maximizing its chosen performance function
or Revenue; i. e., each firm chooses performance function i
If both firms choose Profit in stage I, the result is the standard Cournot-Nash Equilibrium A. If both
firms choose Revenue in stage I, it is straightforward to show that equilibrium quantities and profits in
stage II are
Equilibrium RR
qiRR = 1 a
3
RR
i
= 1 a(a
9
3ci )
Note that since both firms ignore their costs when choosing quantities, each form chooses the same
quantity. Furthermore, the costs do not affect equilibrium quantities and only affect profits.
The last possibility in Stage I is for firm i to choose Revenue and firm j to choose Profit. It is
straightforward to show that in the resulting Stage II equilibrium quantities and profits are
Equilibrium RP
q
RP
i
RP
i
=
1
3
(a + c j )
1 (a 2c )
q RP
j =
j
=
1
9
(a + c j )(a 3ci + c j )
3
RP
j
= 1 (a 2c j )2
9
Note that since firm i commits to ignore its cost when choosing quantity, ci does not affect either firm’s
equilibrium quantity or firm j’s equilibrium profit.
Profits earned in PP, RR, and RP equilibria comprise the payoff matrix of the Stage I game as shown
in Table 2. Each of the four outcomes can be a Nash equilibrium under some combination of cost and
demand parameters. Table 3 summarizes conditions on cost and demand parameters that make each
outcome a Nash equilibrium. Figure 1 plots the corresponding regions in the space of all possible
duopolies parameterized by firms’ costs (c1, c2).
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Strategic commitment to pursue a goal other than profit in a Cournot duopoly (Rtischev)
Table 2. Payoff matrix of the game in which firms may choose to maximize either
Revenue or Profit. Firm 1’s profit appears above firm 2’s profit in each cell. To avoid
fractions, all profits are scaled by a factor of 9.
Profit
Firm 1
Firm 2
(a 2c1 + c2 )2
(a 2c2 + c1 )2
Profit
(a + c2 )(a 3c1 + c2 )
(a 2c2 )2
Revenue
Revenue
(a 2c1 )2
(a + c1 )(a 3c2 + c1 )
a(a 3c1 )
a(a 3c2 )
Table 3. Correspondence between duopoly parameters and management
philosophies chosen in equilibrium
Cost and demand parameters
{(c1 , c2 ) ¦ 1 (a + c1 ) < c2 < 4c1
4
a}
Performance function
chosen in equilibrium
Firm 1
Firm 2
Profit
Profit
{(c1 , c2 ) ¦ c1 < 1 a and c2 < 14 a}
Revenue
Revenue
{(c1 , c2 ) ¦ c2 > max(4c1 a , 1 a)}
Revenue
Profit
{(c1 , c2 ) ¦ c1 > 1 a and c2 < 1 (a + c1 )}
Profit
Revenue
4
4
4
4
Examining the above results leads to the following conclusions about how the possibility of
commitment to pursue revenue instead of profits affects firms’ performance:
1. Both firms maximize profit (region AZC in Figure 1): When both firms’ costs are high, neither firm
makes use of the revenue-maximizing management philosophy. Standard Cournot Equilibrium A
obtains.
2. B
oth firms maximize revenue (region XFB in Figure 1): When both firms’ costs are low, both firms
commit to act as if unit costs were zero and maximize revenue. Compared to standard Cournot
equilibrium A, in equilibrium RR firm 1 and industry as a whole earn less profit. Firm 2’s profit is
{(c1 ,c2 ) ¦ a(c2 2c1 ) > (2c2 c1 )2 } Thus,
also lower except for a limited set of duopolies satisfying .
even though committing to maximize revenue intensifies competition relative to when firms
maximize profit, it is possible for the less efficient firm to benefit. This is because if both firms
commit to maximize revenue, firm 2’s cost disadvantage becomes irrelevant for choosing quantity,
and thus both firms end up on equal footing.
139
c2
c2 = c1
c2 = 4c1 a
Z
c2 = 1 (a + c1 )
2
A
PP
1a
2
q2 = 0
Y
RP
1a F
4
RR
X
E
c2 = 1 (a + c1 )
4
D
C
B
PR
c1
1a
4
*2 0 when both firms
c1 and q　Figure 1 Triangle XYZ is the space of all duopolies such that c　2
maximize profit. Equilibrium management philosophies are denoted by PP (profit maximization by both
firms, region AZC), RR (revenue maximization by both firms, region XFB), RP (revenue-maximization
by firm 1, profit-maximization by firm 2, region FYACB), and PR (profit-maximization by firm 1,
revenue-maximization by firm 2, region BDC).
3. F
irm 1 maximizes revenue, firm 2 profit (region FYACB in Figure 1): When firm 1 is much more
efficient than firm 2, there is a Nash equilibrium in which firm 1 commits to maximize revenue
while firm 2 maximizes profit. Relative to Equilibrium A, in Equilibrium RP firm 1 earns more
profit and firm 2 earns less. Indeed, by adopting the revenue-maximization philosophy, firm 1 may
cause firm 2 to shut down completely; this occurs if c2 > 12 a, which corresponds to region YAE in
1 a + c otherwise it is lower.
Figure 1. Industry profit is higher if c　;
2 >
1
5
4. F
irm 1 maximizes profit, firm 2 revenue (region BDC in Figure 1): For a small subset of parameter
values for which [Revenue, Profit] is an equilibrium, the opposite configuration of philosophies also
constitutes a Nash equilibrium. A coordination problem appears in which each firm prefers to adopt
a philosophy different from that of its rival. Compared to Equilibrium A, in the [Profit, Revenue]
equilibrium firm 2’s profits are higher whereas the profits of firm 1 and the industry as a whole are
lower.
The [Revenue, Profit] equilibrium has also been studied by Blinder (1993), who was interested in
analyzing competition between a revenue-maximizing firm and a profit-maximizing firm. Blinder
concluded that the revenue-maximizer has strategic advantage over the profit-maximizer. However, in
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Strategic commitment to pursue a goal other than profit in a Cournot duopoly (Rtischev)
Blinder’s model firms have identical costs, one firm is exogenously assumed to maximize revenue, and
the other profit. Allowing for cost differences and endogenizing each firm’s decision whether to
maximize profit or revenue, our model identifies parameter ranges within which Blinder’s conclusions
are valid.
5. Discussion
The business press regularly reports about companies pursuing revenue, market share, or some other
performance metric, even to the point of sacrificing profit. The theory we have reviewed suggests one
explanation for why such pursuits makes strategic sense. But because the theory hinges on the firms’
ability to make public and credible commitments to their chosen performance metrics, the explanation is
incomplete without understanding how the firms make such commitments. One possibility is that the
commitments are rooted is the institutional environment within which firms operate. For example, the
law and norms governing firms’ relations with employees and shareholders are such that employee
welfare enters the objective function of many Japanese firms to a greater extent than in the case of
American firms (Aoki, 1988). Starting with this observation, Blinder (1993) showed that a firm which
includes employee welfare along with profit in its performance function essentially becomes a revenuemaximizing entity. Similar to our findings in the RP equilibrium of Section 4, Blinder showed that such
commitment to maximize revenue gives the prototypical Japanese firm a competitive advantage vis-à-vis
the prototypical American firm.
Other ways for a firm to commit to pursue something other than profit have been studied under the
rubric of strategic delegation. Strategic delegation models consider a principal (firm owner) who hires an
agent (manager) to operate the firm. In one strand of strategic delegation literature, the owner hires a
wealth-maximizing manager under an incentive contract that compensates the manager according to
some combination of performance metrics, including those that measure performance relative to
competitors. Taking this approach, Fershtman and Judd (1987) showed that “profit-maximizing owners
will almost never tell their managers to maximize profits.” In another strand of the strategic delegation
literature, the owner selects a manager with certain personality traits, such as compulsion to outdo
competitors (Miller and Pazgal, 2002) or undue optimism about research and development prospects
(Englmaier, 2011). In both strands of strategic delegation literature, commitment to maximize something
other than profit is rooted in the psychology of the manager, who is either rationally maximizing his
private wealth per his incentive contract or irrationally pursuing goals as dictated by his overly rivalrous
or unduly optimistic personality.
The conclusion is that to earn maximum possible profit, a firm does not necessarily have to pursue
maximum profit all the time at every level. A firm which is not overtly pursuing the “bottom line” is not
necessarily guilty of poor management, neglect of shareholder interests, or anticompetitive machinations.
Rather, our theoretical review indicates that quite the opposite is true: strategic deviations from profit
maximization are inherent to the logic of competition in oligopolistic industries. It is a task of strategic
management to discover, commit to, promulgate, and maximize performance metrics which ultimately
yield maximum possible profits, all while taking into account the performance metrics that rivals have
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committed to pursue. The feasibility of a performance metric critically depends on the ability to credibly
commit to use the metric and communicate about it. Contractual, institutional, and psychological bases
of commitment can be of use.
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